7
$\begingroup$

I need to simulate an AR(1) process with the following equation in R:

$$ X_{t} = 5 + 0.5X_{t-1}+Z_t $$

Where $Z_t$ ~ White Noise(0,1) and $T=500$.

I know I should be using the arima.sim function from the forecast package with parameters n = 500 and ar = c(0.5) but I do not know how to account for the $ \delta = 5$ or the fact that $Z_t$ is white noise with mean 0 and sd 1. I have been unable to find clear documentation on how to do this.

$\endgroup$
4
  • $\begingroup$ what is $\delta$? $\endgroup$
    – TPArrow
    Sep 27 '17 at 16:11
  • 1
    $\begingroup$ I vote to leave this question open, as the solution isn't totally programming-related. $\endgroup$
    – Taylor
    Sep 27 '17 at 16:34
  • $\begingroup$ It would help if you explained what $T$ is and where the $\delta$ comes from as otherwise it is unclear, at least to me, what you are asking. $\endgroup$
    – mdewey
    Sep 27 '17 at 17:32
  • $\begingroup$ $T$ is essentially the number of samples, $\delta$ is another component of the AR(1) process as specified in the equation as 5. $\endgroup$
    – sammyj
    Sep 27 '17 at 17:34
8
$\begingroup$

Notice that this is the same model as $$ X_t - 10 = .5(X_{t-1} - 10) + Z_t . $$ $10$ is the mean, while $5$ was the intercept. This means we can add ten to the mean zero series.

In other words, if you define $Y_t = X_t - 10$, then $$ Y_t = .5 Y_{t-1} + Z_t $$ and use arima.sim to simulate that. After you have $\{Y_t\}$, then add $10$ to each value.

Code would look like this:

yt <- arima.sim(list(order=c(1,0,0), ar=.5), n=500)
xt <- yt + 10   

Regarding the standard deviation issue, that is covered in the documentation.

$\endgroup$
1
  • $\begingroup$ Thank you! This is very helpful. I will have to relook into the documentation as I couldn't find where to adjust the mean and standard deviation of the white noise. $\endgroup$
    – sammyj
    Sep 27 '17 at 17:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.